Solve: Half A Number Plus A Third Of Its Consecutive Equals 7

Hey there, math enthusiasts! Ever stumbled upon a word problem that felt like a riddle wrapped in an enigma? Well, today we're diving headfirst into one of those intriguing puzzles. Our mission, should we choose to accept it, is to crack this numerical conundrum: "The sum of half a number and one-third of its consecutive number is 7. What's the number?" Sounds like a fun challenge, right? Let's put on our thinking caps and get started!

Setting Up the Equation: The Key to the Kingdom

In the realm of mathematics, word problems are like secret codes waiting to be deciphered. The first step in unlocking the solution is translating the words into a mathematical equation. This is where our algebraic prowess comes into play. So, let’s break down the given statement piece by piece.

"Half a number"—this phrase immediately suggests division. If we represent our unknown number with the variable 'x', then half of that number is simply x/2. Easy peasy, right?

Now, let's tackle the next part: "the third part of its consecutive number." A consecutive number is just the number that comes right after our original number. So, if our number is 'x', its consecutive number is 'x + 1'. Taking one-third of this consecutive number means we have (x + 1) / 3. We're on a roll!

The problem tells us that the sum of these two quantities is equal to 7. So, we can now construct our equation:

x/2 + (x + 1)/3 = 7

This equation is the heart of our problem. It's the key that will unlock the mystery number. All we need to do now is solve for 'x'. But how do we do that? Fear not, because we're about to embark on a step-by-step journey through the world of algebraic manipulation.

Solving the Equation: A Step-by-Step Adventure

Solving an equation is like navigating a maze. We need to follow the correct steps to reach the solution. In our case, we have an equation with fractions, which might seem a bit intimidating at first. But don't worry, we've got this! The trick is to eliminate the fractions and work with whole numbers. Here’s how we do it:

Step 1: Finding the Least Common Denominator (LCD)

The first step in banishing those fractions is to find the least common denominator (LCD) of the fractions in our equation. The LCD is the smallest number that is a multiple of both denominators. In our equation, the denominators are 2 and 3. The smallest number that both 2 and 3 divide into is 6. So, our LCD is 6.

Step 2: Multiplying Both Sides by the LCD

Now that we have our LCD, we're going to multiply both sides of the equation by 6. This is a crucial step because it will eliminate the fractions. Remember, whatever we do to one side of the equation, we must do to the other side to keep the equation balanced. So, let's do it:

6 * (x/2 + (x + 1)/3) = 6 * 7

Step 3: Distributing and Simplifying

Next, we distribute the 6 to each term on the left side of the equation:

6 * (x/2) + 6 * ((x + 1)/3) = 42

Now we simplify:

3x + 2(x + 1) = 42

See how the fractions have magically disappeared? We're making progress!

Step 4: Further Simplification

Let's simplify further by distributing the 2 in the second term:

3x + 2x + 2 = 42

Now, combine like terms:

5x + 2 = 42

Step 5: Isolating the Variable

Our goal is to isolate 'x' on one side of the equation. To do this, we first subtract 2 from both sides:

5x = 40

Step 6: Solving for x

Finally, we divide both sides by 5 to solve for 'x':*

x = 8

Eureka! We've found our number! But before we celebrate, let's make sure our solution is correct.

Verifying the Solution: The Final Check

In mathematics, it's always a good idea to verify our solution. This ensures that we haven't made any mistakes along the way. To verify our solution, we substitute x = 8 back into our original equation:

8/2 + (8 + 1)/3 = 7

Let's simplify:

4 + 9/3 = 7

4 + 3 = 7

7 = 7

Our solution checks out! This confirms that our number is indeed 8.

The Answer: The Grand Finale

So, after our mathematical journey, we've arrived at the answer. The number we were searching for is 8. Isn't it satisfying when a puzzle comes together? This problem illustrates the power of algebra in solving real-world problems. By translating words into equations, we can unlock solutions that might otherwise remain hidden.

Real-World Applications: Math in Action

You might be wondering, "Where would I ever use this kind of math in the real world?" Well, the truth is, algebra is all around us! It's used in everything from engineering and computer science to economics and finance. Whenever we need to solve for an unknown quantity, algebra is our trusty tool.

For example, let's say you're planning a road trip. You know the distance you want to travel and the average speed you'll be driving. You can use algebra to calculate how long the trip will take. Or, imagine you're trying to budget your monthly expenses. You can use algebra to figure out how much you can spend on different categories while staying within your budget.

The applications are endless! Math is not just about numbers and equations; it's a way of thinking and problem-solving that can be applied to countless situations.

Conclusion: The Beauty of Mathematical Puzzles

And there you have it, folks! We've successfully cracked the code and found the number that satisfies our equation. Hopefully, this has demystified word problems for you and shown you that they're not as scary as they might seem. Math is like a game, and each problem is a new level to conquer. So, keep practicing, keep exploring, and most importantly, keep having fun with it!

Remember, every problem is an opportunity to learn something new. Whether it's mastering a new algebraic technique or understanding a real-world application, math has something to offer everyone. So, embrace the challenge, and who knows? You might just discover a hidden talent for numbers!